Any system that models the world under finite representational capacity must compress; any compression entails a prior; and the prior is the system's bias. What has not been established is whether uncertainty participates in the dynamics governing future behavior, or merely describes the output distribution without consequence. We introduce a structural distinction between descriptive uncertainty, which does not recursively modulate the system's policy, and regulatory uncertainty, which directly enters the optimization landscape and drives persistent adaptive restructuring. We prove formally that current transformer architectures are confined to descriptive uncertainty at inference. We ground this in thermodynamics via Landauer's principle: for uncertainty to be regulatory, epistemic error must cost real energy; in a decoupled system, hallucinations and correct derivations dissipate identical energy. We test this empirically across three locally-deployed language models (3B, 8B, 70B parameters). Token-level Shannon entropy is statistically invariant across tasks spanning pattern retrieval, causal operator application, and out-of-distribution causal generalization in all three models (all pairwise p >= 0.568; within-model ranges 0.011-0.028 nats), while task accuracy varies substantially across the same conditions (0%-100%). Entropy and accuracy are orthogonal. The decoupling is scale-invariant: larger models achieve higher accuracy but identical entropy flatness. This structural incapacity is not resolvable by additional parameters or training data. Genuine epistemic grounding requires physical coupling between thermodynamic substrate state and information processing cost.
The distinction between aleatoric and epistemic uncertainty has received considerable attention in machine learning research, mainly in the context of supervised learning but also in other settings such as generative modeling. In this paper, we offer a machine learning perspective on uncertainty modeling for dynamical systems, which has been studied much less so far. In particular, we ask: what uncertainties do we need for dynamical systems? We discuss sources of uncertainty, clarify their nature (aleatoric or epistemic), and consider how the objectives of representing and quantifying uncertainty vary across different tasks.
Uncertainty quantification is crucial in machine learning, yet most (axiomatic) studies of uncertainty measures focus on classification, leaving a gap in regression settings with limited formal justification and evaluations. In this work, we provide a formal way of representing uncertainty in continuous space, using a general parametric formulation, allowing for tractable analysis and evaluation of uncertainty measures. Within this framework, we propose a set of axioms that enable rigorous assessment of total, aleatoric, and epistemic uncertainty measures. Together, this allows for a theoretical examination of uncertainty measures and their corresponding properties. As a specific example, we compare the widely used entropy- and variance-based measures with respect to established predictive models and analyze their limitations and challenges in uncertainty quantification. Our work provides a principled way to understand and develop uncertainty measures in supervised regression, offering theoretical insights and practical guidelines for reliable uncertainty assessment.
In autoregressive large language models (LLMs), temporal straightening offers an account of how the next-token prediction objective shapes representations. Models learn to progressively straighten the representational trajectory of input sequences across layers, potentially facilitating next-token prediction via linear extrapolation. However, a direct link between this trajectory and token-level behavior has been missing. We provide such a link by relating contextual curvature-a geometric measure of how sharply the representational trajectory bends over recent context-to next-token entropy. Across two models (GPT-2 XL and Pythia-2.8B), contextual curvature is correlated with entropy, and this relationship emerges during training. Perturbation experiments reveal selective dependence: manipulating curvature through trajectory-aligned interventions reliably modulates entropy, while geometrically misaligned perturbations have no effect. Finally, regularizing representations to be straighter during training modestly reduces token-level entropy without degrading validation loss. These results identify trajectory curvature as a task-aligned representational feature that influences behavioral uncertainty in LLMs.